A sequential ensemble approach to epidemic modeling: Combining Hawkes and SEIR models using SMC$^2$

TL;DR

This paper tackles real-time epidemic forecasting under model and parameter uncertainty by proposing a sequential ensemble framework that combines a discrete-time Hawkes process (DTHP) and a stochastic SEIR model using SMC$^2$. At each step, model outputs are weighted by posterior model probabilities computed via a sliding-window incremental marginal likelihood, producing a model-averaged predictive distribution for incidence and the time-varying reproduction number $R_t$, while propagating uncertainty in both states and parameters. Across synthetic scenarios and Irish influenza and COVID-19 data, the ensemble often yields more accurate and stable forecasts than either model alone and provides calibrated predictive intervals. The approach offers practical value for adaptive epidemic surveillance and public health decision-making, with potential extensions to handle abrupt regime shifts and additional data streams.

Abstract

This paper proposes a sequential ensemble methodology for epidemic modeling that integrates discrete-time Hawkes processes (DTHP) and Susceptible-Exposed-Infectious-Removed (SEIR) models. Motivated by the need for accurate and reliable epidemic forecasts to inform timely public health interventions, we develop a flexible model averaging (MA) framework using Sequential Monte Carlo Squared. While generating estimates from each model individually, our approach dynamically assigns them weights based on their incrementally estimated marginal likelihoods, accounting for both model and parameter uncertainty, to produce a single ensemble estimate. We assess the methodology through simulation studies mimicking abrupt changes in epidemic dynamics, followed by an application to the Irish influenza and COVID-19 epidemics. Our results show that combining the two models can improve both estimates of the infection trajectory and reproduction number compared to using either model alone. Moreover, the MA consistently produces more stable and informative estimates of the time-varying reproduction number, with credible intervals that provide a realistic assessment of uncertainty. These features are particularly useful when epidemic dynamics change rapidly, enabling more reliable short-term forecasts and timely public health decisions. This research contributes to pandemic preparedness by enhancing forecast reliability and supporting more informed public health responses.

Figures (12)

• Figure 1: Simulated daily incidence and time-varying reproduction number for Scenario A. The black/brown stars correspond to the observed incidence data, and the true underlying $R_t$ is shown as a solid brown line. The solid lines represent the posterior mean estimates of the incidence and $R_t$ obtained from the DTHP (orange), SEIR (blue), and MA (green) approaches, with shaded areas indicating the associated 95% credible intervals. The vertical black dashed line marks the start of the forecasting period.
• Figure 2: Simulated daily incidence and time-varying reproduction number for Scenario B. The black/brown stars correspond to the observed incidence data, and the true underlying $R_t$ is shown as a solid brown line. The solid lines represent the posterior mean estimates of the incidence and $R_t$ obtained from the DTHP (orange), SEIR (blue), and MA (green) approaches, with shaded areas indicating the associated 95% credible intervals. The vertical black dashed line marks the start of the forecasting period.
• Figure 3: Simulated daily incidence and time-varying reproduction number for Scenario C. The black/brown stars correspond to the observed incidence data, and the true underlying $R_t$ is shown as a solid brown line. The solid lines represent the posterior mean estimates of the incidence and $R_t$ obtained from the DTHP (orange), SEIR (blue), and MA (green) approaches, with shaded areas indicating the associated 95% credible intervals. The vertical black dashed line marks the start of the forecasting period.
• Figure 4: weekly incidence data from the Influenza pandemic and corresponding filtering estimate of the time-varying reproduction number for the DTHP and SEIRS models. The black/brown starts correspond to the observed data for incidence. The solid lines represent the posterior mean estimates of the incidence and $R_t$ obtained from the DTHP (orange), and SEIRS (blue) approaches, with shaded areas indicating the associated 95% credible intervals. The vertical black dashed line marks the start of the forecasting period, and the light gray area highlights the forecast window. Summer periods are indicated by a light pink area.
• Figure 5: Daily case incidence from the COVID-19 and corresponding filtering estimate of the time-varying reproduction number. The black/brown starts correspond to the observed data for incidence. The solid lines represent the posterior mean estimates of the incidence and $R_t$ obtained from the DTHP (orange), SEIR (blue), and MA (green) approaches, with shaded areas indicating the associated 95% credible intervals. Vertical red dashed lines mark the start dates of major mitigation measures, the vertical black dashed line indicates the beginning of the forecasting period, and the light gray area denotes the forecast window.